The following question comes from my studying of Alain Connes's paper [Trace Formula in Noncommutative Geometry and the Zeros of the Riemann Zeta Function][1]. In it, on p. 11, Connes notes that if $K$ is ***local*** the additive Haar measure $dx$ on $(K,+)$ and the multiplicative Haar measure $d^{*}x$ on $K^{*}$ are related by the equation

\begin{equation*}
dx=|x|d^{*}x\text{.}
\end{equation*}

So far, so good. Connes now continues by making the claim, which he seems to take for granted as trivial, that if $K$ is ***global***, one has

\begin{equation*}
dx=\lim_{\epsilon\longrightarrow 0}\epsilon|x|^{1+\epsilon}d^{*}x\text{.}
\end{equation*}

I have no clue why that should be the case. It is very possible that I may not be seeing the forest for all the trees here. Can anyone help?

  [1]: https://arxiv.org/abs/math/9811068